In complex geometry, an Inoue surface is any of several complex surfaces of Kodaira class VII.
They are named after Masahisa Inoue, who gave the first non-trivial examples of Kodaira class VII surfaces in 1974.
[1] The Inoue surfaces are not Kähler manifolds.
Inoue introduced three families of surfaces, S0, S+ and S−, which are compact quotients of
(a product of a complex plane by a half-plane).
by a solvable discrete group which acts holomorphically on
The solvmanifold surfaces constructed by Inoue all have second Betti number
These surfaces are of Kodaira class VII, which means that they have
It was proven by Bogomolov,[2] Li–Yau[3] and Teleman[4] that any surface of class VII with
is a Hopf surface or an Inoue-type solvmanifold.
These surfaces have no meromorphic functions and no curves.
The Inoue surfaces are constructed explicitly as follows.
[5] Let φ be an integer 3 × 3 matrix, with two complex eigenvalues
is called Inoue surface of type
The Inoue surface of type S0 is determined by the choice of an integer matrix φ, constrained as above.
be the group of upper triangular matrices The quotient of
as a matrix with two positive real eigenvalues a, b, and ab = 1.
as φ. Identifying the group of upper triangular matrices with
The same argument as for Inoue surfaces of type
is called Inoue surface of type
are defined in the same way as for S+, but two eigenvalues a, b of φ acting on
have opposite sign and satisfy ab = −1.
Since a square of such an endomorphism defines an Inoue surface of type S+, an Inoue surface of type S− has an unramified double cover of type S+.
Parabolic and hyperbolic Inoue surfaces are Kodaira class VII surfaces defined by Iku Nakamura in 1984.
These surfaces have positive second Betti number.
They have spherical shells, and can be deformed into a blown-up Hopf surface.
Parabolic Inoue surfaces contain a cycle of rational curves with 0 self-intersection and an elliptic curve.
They are a particular case of Enoki surfaces which have a cycle of rational curves with zero self-intersection but without elliptic curve.
Half-Inoue surfaces contain a cycle C of rational curves and are a quotient of a hyperbolic Inoue surface with two cycles of rational curves.
All these surfaces may be constructed by non invertible contractions.